Question

The probability that a man will live $10$ more years, is $\frac{1}{4}$ and the probability that his wife will live $10$ more years, is $\frac{1}{3}$. Then, what is the probability that neither will be alive in $10$ years?

• A. $\frac{1}{2}$
• B. $\frac{3}{7}$
• C. $\frac{2}{3}$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $\frac{1}{2}$

let M be the event that the man will live 10 years more

let W be the event that the wife will live 10 years more

Given, $P\left(M\right)=\frac{1}{4}\Rightarrow P\left(\overline{M}\right)=1-\frac{1}{4}=\frac{3}{4}$
and $P\left(W\right)=\frac{1}{3}\Rightarrow P\left(\overline{W}\right)=1-\frac{1}{3}=\frac{2}{3}$
Here, both events are independent, so the required probability
$=P(\overline{W}\cap \overline{M})$
$=P\left(\overline{W}\right)\cdot P\left(\overline{M}\right)$
$=\frac{3}{4}\times \frac{2}{3}$
$=\frac{1}{2}$